Evolutionary Systems Representations Based on Later Time Samples and Applications to PDEs
Yeon Hyang Kim, Roza Aceska · East Asian Journal on Applied Mathematics · 2020
Let $\{ƒ_n\}^∞_{n=1}$ be a basis for $L_2([0, 1])$ and $\{g_n\}^∞_{n=1}$ be a system of functions of controlled decay on [0,∞). Considering a function u(x, t) that can be the represented in the form\[ u(x,t) = \sum_{n=1}^{\infty} a_n f_n(x) g_n(t), \]where an ∈ R, x ∈ [0, 1] and t ∈ [0,∞), we investigate whether the function ƒ(x) = u(x, 0) can be approximated, in a reasonable sense, by using data u(x0, t1), u(x0, t2),..., u(x0, tN). A mathematical framework and efficient computational schemes are developed to determine approximate solutions for various classes of partial differential equations via sampled data by first establishing a near-best approximation of ƒ .